Long-Term Stability of the HR 8799 Planetary System Without Resonant Lock

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Long-term stability of the HR 8799 planetary system without resonant lock

Götberg, Y.; Davies, M.B.; Mustill, A.J.; Johansen, A.; Church, R.P. DOI 10.1051/0004-6361/201526309 Publication date 2016 Document Version Final published version Published in Astronomy & Astrophysics

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Citation for published version (APA): Götberg, Y., Davies, M. B., Mustill, A. J., Johansen, A., & Church, R. P. (2016). Long-term stability of the HR 8799 planetary system without resonant lock. Astronomy & Astrophysics, 592, [A147]. https://doi.org/10.1051/0004-6361/201526309

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:28 Sep 2021 A&A 592, A147 (2016) DOI: 10.1051/0004-6361/201526309 Astronomy c ESO 2016 & Astrophysics

Long-term stability of the HR 8799 planetary system without resonant lock

Ylva Götberg1,2, Melvyn B. Davies1, Alexander J. Mustill1, Anders Johansen1, and Ross P. Church1

1 Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, 22100 Lund, Sweden e-mail: [email protected] 2 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

Received 14 April 2015 / Accepted 7 June 2016

ABSTRACT

HR 8799 is a star accompanied by four massive planets on wide orbits. The observed planetary configuration has been shown to be unstable on a timescale much shorter than the estimated age of the system ( 30 Myr) unless the planets are locked into mean motion resonances. This condition is characterised by small-amplitude libration of∼ one or more resonant angles that stabilise the system by preventing close encounters. We simulate planetary systems similar to the HR 8799 planetary system, exploring the parameter space in separation between the orbits, planetary masses and distance from the Sun to the star. We find systems that look like HR 8799 and remain stable for longer than the estimated age of HR 8799. None of our systems are forced into resonances. We find, with nominal masses (Mb = 5 MJup and Mc,d,e = 7 MJup) and in a narrow range of orbit separations, that 5 of 100 systems match the observations and lifetime. Considering a broad range of orbit separations, we find 12 of 900 similar systems. The systems survive significantly longer because of their slightly increased initial orbit separations compared to assuming circular orbits from the observed positions. A small increase in separation leads to a significant increase in survival time. The low eccentricity the orbits develop from gravitational interaction is enough for the planets to match the observations. With lower masses, but still comfortably within the estimated planet mass uncertainty, we find 18 of 100 matching and long-lived systems in a narrow orbital separation range. In the broad separation range, we find 82 of 900 matching systems. Our results imply that the planets in the HR 8799 system do not have to be in strong mean motion resonances. We also investigate the future of wide-orbit planetary systems using our HR 8799 analogues. We find that 80% of the systems have two planets left after strong planet-planet scattering and these are on eccentric orbits with semi-major axes of a1 10 AU and a2 30 1000 AU. We speculate that other wide-orbit planetary systems, such as AB Pic and HD 106906, are the remnants∼ of HR 8799 analogues∼ − that underwent close encounters and dynamical instability. Key words. planets and satellites: dynamical evolution and stability

1. Introduction an age of >100 Myr is assumed. In this paper we consider HR 8799 bcde∼ as planets; the case of brown dwarfs is more HR 8799 (also called HD 218396 and HIP 114189) is a nearby +20 closely investigated by Moro-Martín et al.(2010). We assume (39.4 1.1 pc; van Leeuwen 2007), young (30 10 Myr assuming ± − the age 30 Myr in this paper as it is estimated from the Columba member of the Columba association; Marois et al. 2010) A5V association. An age of 30 Myr leads to multi-Jovian masses that, (up to F0V) star with mass 1.5 0.3 M (Gray & Kaye 1999; ± however, are still in the planetary regime. Whether it is true that Gray et al. 2003). HR 8799 has four confirmed, directly imaged HR 8799 is associated with the Columba association could be re- planets (Marois et al. 2008, 2010) and a debris disk (Su et al. vealed by Gaia in the near future (Perryman et al. 2001). Other 2009; Matthews et al. 2014). The planet masses are estimated to uncertainties in system properties include distance (39.4 1.1 pc, be 5 (b) and 7 (cde) MJup (Marois et al. 2010). The on-sky sep- van Leeuwen 2007) and stellar mass (1.5 0.3 M , Gray± & Kaye arations between planets and star are estimated to be 67.9 (b), 1999; Gray et al. 2003). ± 38.0 (c), 24.5 (d), and 14.5 (e) AU. Matthews et al.(2014) estimated the inclination of the system with respect to the plane The formation of the massive HR 8799 bcde planets can be of the sky to be 26 3◦ by observing the outer part of the de- explained by the recently developed models of pebble accretion bris disk and assuming± coplanarity with the planetary orbits. (Lambrechts & Johansen 2012), but the dynamics of the system They estimate the position angle of the inclination to be 64 3◦ has been a puzzle unless resonances are considered. The orbits (G. Kennedy, private communication). Figure1 shows all obser-± of the HR 8799 bcde planets are difficult to constrain as the plan- vations of the planets published at the time of writing. ets have been observed during a small fraction of their orbits (see The estimates of the planetary masses in the HR 8799 sys- Fig.1). In recent work by Pueyo et al.(2015), however, fits are tem depend on the age of the system through cooling models, made to constrain the orbital elements and a variety of solutions indicating higher planet masses with higher age (Baraffe et al. are found, favouring a slightly more eccentric orbit for planet d. 2003). The age of the HR 8799 system is uncertain and has been It is reasonable to believe that the planet orbits are close to circu- estimated using different techniques to range between 30 Myr lar, as with high eccentricity the orbits are likely to cross. How- and 1 Gyr (see Marois et al. 2010, 2008, and references therein). ever, the HR 8799 planetary system appears to be unstable on The planet mass estimates reach the brown dwarf regime when a timescale much shorter than the estimated age of the system

Article published by EDP Sciences A147, page 1 of 14 A&A 592, A147 (2016)

2 2. Simulations 2.1. Orbital parameters 1.5 All planets in all our simulations are analogues of the b 1 Pos. angle HR 8799 bcde planets and they are initiated on circular, helio- 64◦ ± 3◦ c centric orbits. The initial orbit inclination we ﬁnd by drawing a 0.5 random inclination between 0◦ and 5◦ and random longitude of ascending node between 0 and 360◦ (β = 5◦; see Johansen et al. 2012). The initial planet position on the orbit (true anomaly) is 0 e randomised. We assign the mass 1.5 M to the star and the plan-

North ["] ets are given the estimated values of the masses, 5 MJup (b) and −0.5 d 7 MJup (cde) if nothing else is stated (the masses are decreased in simulations 5 and 6). The planets orbit the barycentre of the −1 system, which is not centred on the star as a result of the high planet mass. This means that the planetary orbits have a low ini- −1.5 tial eccentricity as they are initiated on orbits circular around the star. The initial orbital eccentricities vary between different sys- −2 tems as the inclinations, longitude of ascending nodes, and true 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 anomalies are randomised. East ["] We vary the separation between the initial planet orbits (re- garding the planets as test particles) in terms of the mutual Hill Fig. 1. Observations of the HR 8799 planets. The black dashed line radius, which is defined as follows: corresponds to the position angle of the inclination as estimated by !1/3 Matthews et al.(2014) with the 1σ uncertainty shown by the grey rp1 + rp2 Mp1 + Mp2 ± RH = , (1) region. 2 3M? when integrating the orbits of the planets, assumed initially cir- where r signifies star-planet separation in the orbital plane and M cular, seen pole-on and non-resonant (Fabrycky & Murray-Clay mass (Chambers et al. 1996). The indices p1 and p2 refer to the 2010; Reidemeister et al. 2009; Sudol & Haghighipour 2012). two planets and ? refers to the star. We then write the separation A possible and popular explanation to the survival of the between the initial, circular orbits in mutual Hill radii as HR 8799 planetary system is that the planets are locked deep ∆ = (r r )/R , (2) in strong mean motion resonances that can lead to stability p2 − p1 H timescales up to 1 Gyr (e.g. Fabrycky & Murray-Clay 2010; where in this case p2 refers to the outer planet and p1 to the inner Go´zdziewski & Migaszewski 2009; Reidemeister et al. 2009; planet. For a given planetary system there is a critical ∆ at which Moro-Martín et al. 2010; Currie et al. 2011; Soummer et al. in 50% of the cases the system leads to orbit crossing within 2011; Esposito et al. 2013; Go´zdziewski & Migaszewski 2014). a given time (Chambers et al. 1996; Chatterjee et al. 2008), al- These resonant configurations are characterised by small- though for any ∆ there is considerable scatter in the times in amplitude librations of one or more resonant arguments corre- which particular realisations become unstable. sponding to two-, three-, or four-planet mean motion resonances, thus ensuring that systems remain protected for extended periods 2.2. MERCURY6 parameters from close encounters between the planets. Moore & Quillen (2013) find stabilising effects from the outer debris disk, al- We perform our simulations using the MERCURY6 package though this requires a very massive debris disk. (Chambers 1999). We use the Bulirsch-Stoer integrator and a Ja- In this paper we show simulations of planetary systems that cobian coordinate system in the output files. Ejection distance is look like the HR 8799 system, are stable for longer than the es- 11 000 AU and fragmentation from collisions is not allowed. We timated age, and are not stabilised by strong resonant lock. We have redefined a close encounter as when two planets are sepa- create these systems by starting the planetary orbits slightly more rated by less than one mutual Hill radius, defined as in Eq. (1). widely separated than the observed on-sky separation assuming circular orbits. A more widely separated configuration can also 2.3. Simulation parameters look like the HR 8799 system as the orbits are not exactly circular. Larger separation between the planetary orbits results in In all simulations apart from 4f_long we stop the integration a much longer stability timescale, which makes systems remain when a close encounter has occurred or 100 Myr is reached. In stable for longer than the estimated age of HR 8799. This solu- 4f_long we continue running until 100 Myr to investigate the fu- tion is different from previously published explanations, as we ture of multi-planet systems in Sect.5. We use output rates of do not need protective resonant lock or any other stabilising ef- 10, 100 or 1000 yrs. Every simulation contains 100 runs with fects to create a long-lived system. the difference in initial placement on orbits and the inclinations In Sect.2 we describe how we simulate planetary systems. of the orbits. The simulations we carried out are summarised in Sections3 and4 present first results consistent with earlier work Table1 and described in detail in Sects.3 and4. We provide the and later long-lived solutions in agreement with the observed initial conditions of our example system in Table A.1, but we system. In Sect.5 we introduce a concept of evolutionary phases note that the system is so chaotic that a small change in time- in the lives of wide-orbit planetary systems. These phases could step completely alters the lifetime of the system (see Sect. 4.3). be used to better understand the past and future of a wide-orbit A more successful method is to create statistically similar, long- planetary system. We summarise our findings and conclusions in lived systems as explained in this section and described in Sect.4 Sect.6. (see also Table2).

A147, page 2 of 14 Y. Götberg et al.: Long-term stability of the HR 8799 planetary system without resonant lock

Table 1. Initial conditions of the simulations.

Name Separation between orbits [RH] Initial semi-major axis [AU] Comment ∆bc ∆cd ∆de ab ac ad ae 1 4.14 3.02 3.55 67.9 38.0 24.5 14.5 Face-on assumption. (Sect. 3.1)

Observed inclination, i = 26 , position 2 3.47 3.31 3.90 67.9 41.9 25.8 14.5 ◦ angle = 64◦. (Sect. 3.2)

Same initial conditions as system 8 in 2α simulation 2. Diﬀer in output rates and thus timestep. (Sect. 4.3)

Optimal inclination, i = 25 , position 3 3.56 3.56 3.56 68.8 41.9 24.8 14.7 ◦ angle = 42◦. (Sect. 3.3)

Equal initial orbit separation in ∆. 4 Planet e ﬁxed at 14.3 AU. (Sect. 4.1) a 3.6 3.6 3.6 68.1 41.2 24.3 14.3 b 3.65 3.65 3.65 69.7 41.9 24.5 14.3 c 3.7 3.7 3.7 71.2 42.5 24.7 14.3 d 3.75 3.75 3.75 72.9 43.2 24.9 14.3 e 3.8 3.8 3.8 74.6 43.9 25.0 14.3 f 3.85 3.85 3.85 76.3 44.6 25.2 14.3 g 3.9 3.9 3.9 78.1 45.3 25.4 14.3 h 3.95 3.95 3.95 79.9 46.0 25.6 14.3 i 4 4 4 81.8 46.7 25.8 14.3 Initialised in the same way as 4f, but f_long 3.85 3.85 3.85 76.3 44.6 25.2 14.3 not stopping after close encounter. Run time 100 Myr. (Sect.5)

Same initial conditions as system 94 in 4α simulation 4e. Diﬀer in output rates and thus timestep. (Sect. 4.3)

Same initial conditions as 5 simulation 4, but planet masses are 4 (b) and 6 (cde) MJup. (Sect. 4.5) a 3.83 3.79 3.79 68.1 41.2 24.3 14.3 b 3.88 3.84 3.84 69.7 41.9 24.5 14.3 c 3.93 3.90 3.90 71.2 42.5 24.7 14.3 d 3.99 3.95 3.95 72.9 43.2 24.9 14.3 e 4.04 4.00 4.00 74.6 43.9 25.0 14.3 f 4.09 4.05 4.05 76.3 44.6 25.2 14.3 g 4.14 4.11 4.11 78.1 45.3 25.4 14.3 h 4.20 4.16 4.16 79.9 46.0 25.6 14.3 i 4.25 4.21 4.21 81.8 46.7 25.8 14.3

Same initial conditions as 6 simulation 4, but planet masses are 3 (b) and 5 (cde) MJup. (Sect. 4.5) a 4.12 4.03 4.03 68.1 41.2 24.3 14.3 b 4.18 4.08 4.08 69.7 41.9 24.5 14.3 c 4.24 4.14 4.14 71.2 42.5 24.7 14.3 d 4.29 4.20 4.20 72.9 43.2 24.9 14.3 e 4.35 4.25 4.25 74.6 43.9 25.0 14.3 f 4.41 4.31 4.31 76.3 44.6 25.2 14.3 g 4.46 4.36 4.36 78.1 45.3 25.4 14.3 h 4.52 4.42 4.42 79.9 46.0 25.6 14.3 i 4.58 4.47 4.47 81.8 46.7 25.8 14.3

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100 ∆min [RH] 50 4 planets 3.5 80 3 planets 3 2 planets 40 ] 1 planet ◦ 2.5 60 30 Simulation 3 2 Simulation 1 Simulation 2 20 1.5 40 Simulation 2 Simulation 3 Inclination [ 1

Number of systems 10 0.5 20 0 0 0 30 60 90 120 150 180

0 Position angle [◦ E of N] 103 104 105 106 107 108

Time [yrs] Fig. 3. Smallest orbit separation, ∆min, between any planet pair changes with inclination and position angle. We indicate the inclination and po- Fig. 2. Number of planets left in the systems as a function of time. Dot- sition angle of the HR 8799 system measured by Matthews et al.(2014) ted line corresponds to the face-on assumption (simulation 1), solid line with a white square (simulation 2), which is i = 26 3◦ and position ± to the inclination inferred from observations (simulation 2), and dashed angle 64 3◦. The largest ∆ occurs at i = 25◦ and position angle 42◦, ± min line to the largest separation achieved from inclination (simulation 3). when ∆min = 3.56 RH. This value is indicated with a ﬁlled, white circle See Table1 and Sect.3 for descriptions about the simulations. The age (simulation 3). This plot is based on the observations from Marois et al. +20 estimate of HR 8799 (30 10 Myr) is indicated with a solid line with the (2008, 2010). grey region corresponding− to the uncertainty.

with the longest-lived system surviving 100 times longer than Throughout the paper we compare our simulations to the ob- the median. servations of Currie et al.(2011) from 8 October 2009, as these observations have low measurement uncertainties and all four planets are observed simultaneously. 3.2. Simulation 2: inclined system according to observations The disk of the HR 8799 system is observed to be inclined by = 3. HR 8799 – unstable as observed i 26◦ compared to the plane of the sky and at a position angle of 64 E of N (64 3 , best fit; G. Kennedy, priv. comm.; ◦ ± ◦ Below we describe in more detail the three simulations 1, 2, and Matthews et al. 2014). Here we assume that the planets and disk 3 seen in Table1. We use the term stability timescale as the are mutually coplanar. These inclination parameters we use also timescale for which the systems in a simulation have not had later in simulations 4, 5, and 6. The observed, on-sky separations any close encounters. This timescale we find is roughly the same between planets and star are then 67.9 (b), 41.9 (c), 25.8 (d), as the time it takes for the systems to lose a planet. and 14.5 (e) AU, which means that planets c and d are located In Fig.2 the number of systems with 4, 3, 2, and 1 planets slightly further away from the star compared to the face-on case in simulations 1, 2, and 3 is shown as a function of time. The (simulation 1). The initial separations between the orbits are then black line indicates the estimated age of HR 8799. The figure ∆ = 3.47 (b-c), 3.31 (c-d), and 3.90 (d-e) RH. We refer to this shows that none of the systems in simulations 1, 2, or 3 still have simulation as simulation 2 (see Table1, solid lines in Fig.2 and four planets when the age of HR 8799 is reached. The figure also white square marker in Fig.3). shows that 80% of the systems have two planets left and 20% The median time during which the systems have not had any have one planet∼ left after 100 Myr. ∼ close encounter is in this case 0.094 Myr, while there are no systems that still have four planets after 9.39 Myr. Compared to simulation 1 the stability timescale is similar and it remains too 3.1. Simulation 1: face-on assumption short to explain the current state of HR 8799. For simplicity, we begin by assuming the HR 8799 system is seen entirely face-on, i.e. inclination i = 0◦ compared to the 3.3. Simulation 3: optimal separation from inclination plane of the sky. We use the observed on-sky star-planet separations of 67.9 (b), 38.0 (c), 24.5 (d), and 14.5 (e) AU as ini- Assume we did not have any system inclination estimate. Which tial star-planet separations. The initial separations between the inclination and position angle would then give the largest plan- orbits are then ∆ = 4.14 (b-c), 3.02 (c-d) and 3.55 (d-e) RH. etary orbit spacing and thus the longest stability timescale? In The dynamics of the face-on assumption has been tested earlier Fig.3 the minimum separation between any two planetary orbits by Fabrycky & Murray-Clay(2010) in the three-planet case and in the system is given for sets of inclination and position angle Marois et al.(2010) with all four planets. We confirm their re- assuming circular orbits centred on the observed mean (using sults by finding a median time at which the planetary systems the observations of Marois et al. 2008, 2010). The largest ∆min have the first close encounter of 0.10 Myr and there is no plan- is found with an inclination of i = 25◦ and a position angle of etary system without close encounters for longer than 9.4 Myr. 42◦. The separation between the orbits are then ∆ = 3.56 RH for We refer to this simulation as simulation 1, see Table1 and dot- all planet pairs, which translates into star-planet separations of ted lines in Fig.2. Although none of our systems live for the 68.8 (b), 41.9 (c), 24.8 (d), and 14.7 (e) AU. We use these opti- age of HR 8799, we point out here the large range of lifetimes mal separation parameters in simulation 3 (see Table1, dashed

A147, page 4 of 14 Y. Götberg et al.: Long-term stability of the HR 8799 planetary system without resonant lock

Table 2. Observed wide-orbit planets around stars.

System Age Spectral type Planets rproj Mp Predicted phase Ref. [Myr] [AU] [MJup] +3 1RXS J1609 5 M0 1V b 330 8 2 B/C 1, 2 AB Pic 30 K2V± b ∼260 13− 14 C 3 ∼ +8 ∼ − β Pic 12 4 A6V b 8 15 9 3 A/C 4, 5 − − ± Fomalhaut 440 A3V b∗ 119 <3 B/C 6, 7 +350 ∼ ∼ +4.5 GJ 504 160 60 G0V b 43.5 4.0 1.0 C 8 HD 106906 13 − 2 F5V b 650 11 − 2 B/C 9, 10 HD 95086 10 ±17 A8V b 56∼.4 0.7 5 ±2 A/B/C 11, 12 +−20 ± +±3 HR 8799 30 10 A5V to F0V b 68 7 2 A 13, 14, 15, 16 − +−3 c 38 7 2 13, 14, 15 +−3 d 24 7 2 13, 14, 15 e 15 5− 2 14, 15 +20 ± +2.0 κ And 30 10 B9IV b 55 2 12.8 1.0 A/C 17, 18 − ± − ( ) Notes. ∗ Note news of Fomalhaut b being dust cloud from collision with asteroid belt (Kennedy & Wyatt 2011; Kenyon et al. 2014). References. 1 – Bowler et al.(2014); 2 – Lafrenière et al.(2010); 3 – Chauvin et al.(2005); 4 – Lagrange et al.(2010); 5 – Zuckerman et al. (2001); 6 – Mamajek(2012); 7 – Kalas et al.(2008); 8 – Kuzuhara et al.(2013); 9 – Bailey et al.(2014); 10 – Pecaut et al.(2012); 11 – Rameau et al.(2013); 12 – Galicher et al.(2014); 13 – Marois et al.(2008); 14 – Marois et al.(2010); 15 – Currie et al.(2011); 16 – Gray & Kaye (1999); 17 – Carson et al.(2013); 18 – Wu et al.(2011). lines in Fig.2 and white circle in Fig.3). The median time we analyse the simulations in terms of resonant behaviour. the systems have not had any close encounter is 0.57 Myr and Section 4.5 describes simulations 5 and 6, which consider lower all systems have had close encounters after 6.85 Myr. Figure2 planet masses that are still within the estimated range. shows that an increase of 0.25 RH in ∆min results in a factor of 2 longer stability timescale (compare medians of simulation 2 ∼ with simulation 3). 4.1. Simulation 4: wider initial orbital separation Preferred estimates of the HR 8799 system inclination from a dynamical point of view lie around 20 30 (Reidemeister et al. − ◦ We test a wider-orbit assumption by simulating systems with 2009; Sudol & Haghighipour 2012), while astroseismological the following initial equal separations between the planet orbits: measurements favour inclinations higher than 40◦ assuming that ∆ = 3.6, 3.65, 3.7, 3.75, 3.8, 3.85, 3.9, 3.95, and 4 RH. We place the planetary orbits are aligned with the stellar spin (Wright et al. planet e at an initial star-planet separation of 14.3 AU, which 2011). then fixes the initial semi-major axes of the other planets. The None of simulations 1, 2 or 3 manage to explain the existence planet masses are set to the nominal masses M = 5 MJup (b) and of the HR 8799 system as they simply fall apart before the age 7 MJup (cde) (Marois et al. 2010). These simulations are called of the system is reached. simulation 4a–i; see Table1. Simulation 4 thus contains 900 runs compared to simulations 1, 2, and 3, which only contain 100 runs each. It makes sense to talk about simulations 4a–i for the ini- 4. A stable solution without resonant lock tialisation of the runs, but not in terms of the architecture of the outcome. The measured minimum orbit separation ∆ for each We find long-lived, simulated systems that look like HR 8799 min system ranges between 3.6 and 4 R in almost each of simula- during some time in their evolution. These systems are created H tions 4a–i, depending on the orbital eccentricities. Therefore we with initially wider separation between the orbits than calculated bin the simulations 4 with ∆ , regardless of the initial set-up. from observations assuming circular orbits. This means that we min We make nine bins with each 100 runs. These bins contain differ- assume the planets are not moving on completely circular orbits. ent ranges of orbit separations (∆ ) and thus represent slightly Our simulated planetary orbits have a low eccentricity of e . min different architectures of the planetary system. 0.05 from the beginning of the integration (for description on how this is set up see Sect. 2.1). The eccentricity enables planets Figure4 shows the time during which the systems in the sim- on more widely separated orbits to look like HR 8799 during ulation 4 bins have not had any close encounter. The figure shows parts of the orbits. A wider separation between the orbits in a the trend of stability timescale with minimum orbit separation, system can make the system survive longer than the estimated ∆min. Compared to the systems in Fig.2 a substantial fraction age of HR 8799 as the stability timescale is strongly dependent of systems in Fig.4 have not experienced any close encounter on orbital separation (Davies et al. 2014). at the estimated age of HR 8799 ( 30 Myr). Chatterjee et al. In this section we present our simulations 4, 5, and 6 (see (2008) find that a larger number of mutual∼ Hill radii are needed Table1), which all contain long-lived systems that look like to keep their planetary systems stable compared to what we find. HR 8799. In Sect. 4.1 we describe simulation 4 and show an For various reasons (e.g. our higher planet masses, lower ini- example of a long-lived system that seems to fit the observations tial eccentricities, and different stability criterion), we consider of HR 8799. In Sect. 4.2 we go into detail on how we determine both results valid and our results not contradicting the results of if a simulated system fits the observations. Section 4.3 presents Chatterjee et al.(2008). We see that at any given separation there a way of finding even more well-fitting systems. In Sect. 4.4 is a wide range of stability times, a fact to which we return later.

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4 3 12 100 the observations would be (2 10− ) = 8 10− for a very well-fitting system (fixing× for example b to× always be at the observed position angle). To catch one output where all 80 Simulation bin, ∆min 11 h i planets fit simultaneously would therefore require 10 out- 3.47 RH puts – a number about three orders of magnitude higher∼ than 3.52 R 60 H our output rate. As during the simulation the planets are at 3.57 RH some point located at exactly the observed position angle, 3.61 RH but we do not catch that moment, it is validated to move the 40 3.65 RH planets along their orbits in this way to really measure how 3.71 RH close they come to the observations. 3.76 RH 2. Draw one million random star-planet separations from the 20 distributions created in step number 1 for each planet. This 3.81 RH is carried out to create a new distribution of star-planet sepa- 3.89 RH

Number of runs with close encounters rations with a method called bootstrapping with resampling. 0 103 104 105 106 107 108 3. To simulate the uncertainty of the observation, draw a normally distributed error to each star-planet separation in the Time [yrs] bootstrapped distribution using the standard deviation of the observations. We show these distributions for the simulated Fig. 4. Number of systems without close encounters shown with time. The different coloured lines correspond to bins of each 100 systems in planets in system 94, simulation 4e in Fig.7. simulation 4, sorted in measured minimum orbit separations (∆min). The 4. Measure for each planet the quantile for the observed star- legend indicates the bins and their systems’ mean value of ∆min. The planet separation in the created distribution. The quantile is black line and the grey zone show the age estimate with uncertainty of the fraction of outputs residing to the left of the observed HR 8799. star-planet separation. The star-planet separations are indicated with black, dark grey, and light grey lines in Fig.7 for + + σ Do the surviving systems in simulation 4 look like HR 8799? Earth-HR 8799 distances of 39.4 pc, then 1 and 2 on the In Fig.5 we compare a system from simulation 2 (left panel, sys- distance estimate. tem 8) with a system from simulation 4e (right panel, system 94). Both runs seem to fit the observations, but the system from sim- The quantiles are the measures for how well the observed planet ulation 4 survives over 45 Myr because of the slightly more position and the simulated planet positions fit. Quantiles close to widely separated planetary orbits. We want to stress that sys- 0.5 would indicate a very good fit. We assess systems with quantiles between 10 4 and 1 10 4 as systems fitting to the observa- tem 94, simulation 4e is not a special system, but seems normal − − − compared to the other systems in simulation 4. As we explain tions. A low limit for the quantiles means that we allow systems in Sect. 4.2 we find 12 systems fitting the observations in sim- that rarely are found with the architecture of the HR 8799 system ulation 4. If we consider a distance to the HR 8799 system 1σ to also count as fitting systems. Our example system, system 94, away from the mean of the estimate (39.4 + 1.1 pc = 40.5 pc) we simulation 4e actually has a lower number for the quantile for almost double the number to 22 fitting systems in simulation 4. planet b and does not count as a fitting system. How we determine if a system fits the observations is discussed A visualisation of the assessment procedure can be seen in detail in Sect. 4.2. in Fig.7 where we show, for system 94, simulation 4e, the To demonstrate the similarity of our example system, system star-planet distributions plotted together with the star-planet 94, to the observed HR 8799 system, we show a snapshot of the distances inferred from the Earth-HR 8799 distances 39.4, 40.5, system at 37.6 Myr in Fig.6. and 41.6 pc (corresponding to the mean of the estimate, then one and two standard deviations away from the mean). The quantiles for system 94, simulation 4e are qe = 0.89, qd = 0.89, 5 4.2. Determining whether a system fits the observations qc = 0.02 and qb = 8.7 10− using the mean of the Earth- HR 8799 distance estimate.× If we consider the distance one stan- In this section we describe in detail how we determine how dard deviation away from the estimate (40.5 pc to HR 8799), each simulated system fits the observations. We have chosen to however, we find the quantiles q = 0.98, q = 0.99, q = 0.16 compare the simulated systems with the observations taken on e d c and q = 0.01. At this distance we consider the system 94, simu- 8 October 2009 by Currie et al.(2011) because the planets are b lation 4e to fit the observations well. To not miss systems such as observed simultaneously during this observation and the mea- system 94, simulation 4e in our analysis, we consider two values surement uncertainties are comparably small. of the Earth-HR 8799 mean of the estimate distance (39.4 pc) We start by checking whether the system has a longer sta- and the distance one standard deviation away from the mean of bility timescale than the estimated age of HR 8799 (30 Myr, the estimate (40.5 pc). Marois et al. 2010). If that is the case, we proceed with checking whether the system looks like HR 8799. We check whether the We summarise our findings of simulation 4 in the top two architecture of the simulated system is similar to the architecture panels of Fig.8. Again we have binned the simulation 4 with of HR 8799 in the following way: measured minimum orbit separation, ∆min, 100 systems in each bin. This makes in total 9 bins and 900 systems. We plot in light 1. For every output, move each planet along its orbit to the blue the fraction of the systems in each bin that have a stability observed position angle. This is done because it is highly timescale longer than the estimated age of HR 8799. In dark unlikely that we find a single output where all four planets blue we show the fraction of systems in each bin that fit the happen to be located within the uncertainties of the obser- observations of the HR 8799 system according to the above de- vations. The space the observations take on compared to the scribed method. The grey bars show the fraction of systems in 4 orbit is about 0.04 AU/200 AU = 2 10− . The chance of each bin that fulfil both criteria and thus fit the HR 8799 system. then catching a point in time when all× four planets are within We find for the nominal Earth-HR 8799 distance (39.4 pc), in

A147, page 6 of 14 Y. Götberg et al.: Long-term stability of the HR 8799 planetary system without resonant lock

System 8, simulation 2 System 94, simulation 4e

2 tce= 1.6 Myr 2 tce= 45.4 Myr

1 1

0 0 North ["] North ["]

1 1 − −

2 2 − − 2 1 0 1 2 2 1 0 1 2 − − − − East ["] East ["] (a) System 8 in simulation 2 shown together with the observa- (b) System 94 in simulation 4e shown together with the obser- tions of the HR 8799 planetary system. The system seems to fit vations of the HR 8799 planetary system. This system seems to the observations well, but suffers a close encounter at an age of fit the observations well until a close encounter occurs between 1.6 Myr between planet d and e. In the figure we show the posi- planets d and e at an age of 45.4 Myr. We use this system as an tions of the simulated planets every 10 years (black dots). example of a long-lived system without strong resonances that also look like HR 8799. In the figure we show the positions of the simulated planets every 100 years (black dots).

Fig. 5. Comparison between a system in simulation 2 and a system in simulation 4e. Both systems seem to ﬁt the observed positions of the planets, but only the system in simulation 4e is stable for longer than the estimated age of HR 8799. This is an on-sky projection of the simulations assuming a distance of 39.4 pc (van Leeuwen 2007) and inclination 26◦ (position angle 64◦, Matthews et al. 2014). The axis around which the system is inclined is indicated with a dashed line. The position of the star is indicated with an asterisk.

2 find 12 of 900 systems that fit the observations (see Fig.8a). For the Earth-HR 8799 distance of 40.5 pc we find, in the best Time: 37.6 Myr 1.5 case (where ∆min = 3.74 3.79) 8 systems of 100 fitting the observations and 22 of 900− when including all orbit separation ranges (see Fig.8b). 1 In simulation 4 we create many varieties of architectures in the planetary systems that they would fit many ranges of inclina- 0.5 tions and position angles. Therefore we are not concerned by the exact values of the inclination and position angle that we apply 0 (from Matthews et al. 2014). North ["] 0.5 − 4.3. Simulations 2α and 4α: verification 1 − System 94, simulation 4e (see e.g. Fig. 5b) is not a special sys- Currie et al. 2011 tem, but there are many evolutions leading to systems that fit the 1.5 observations of HR 8799. To show this, we simulate 100 copies − System 94, simulation 4e of system 94, simulation 4e with the only difference being the 2 simulation output rate, which changes the timestep slightly. This − 2 1.5 1 0.5 0 0.5 1 1.5 2 − − − − very small timestep variation changes the fate of the systems in East ["] 106 years because they are chaotic. The systems have similar ∼architecture, but with different evolution. We call them simula- Fig. 6. System 94, simulation 4e at 37.6 Myr (small black dots) com- tion 4α. pared to the observations of Currie et al.(2011) (coloured, larger dots). The axis around which the system is inclined is shown with a dashed Because our simulated systems are very chaotic it is likely line (position angle 64◦ and inclination 26◦, Matthews et al. 2014). The that an integration made with our exact initial conditions would position of the star is indicated with an asterisk. evolve differently from that which we present here. A small change in timestep alters the evolution on a timecale of 106 yrs and our interesting systems have a stability timescale much the best case (where ∆min = 3.64 3.67) 5 systems of 100 that longer than that. Therefore, we note that our initial condi- fit the observations. In the entire range− of orbit separations, we tions in Table A.1 are likely to not reproduce our system 94,

A147, page 7 of 14 A&A 592, A147 (2016)

e d c b

d⋆ = 39.4 pc d⋆ = 39.4+1.1 pc d⋆ = 39.4+2.2 pc

10 12 14 16 18 18 20 22 24 26 28 30 34 36 38 40 42 44 46 65 70 75 80 85 On-sky projected separation [AU]

Fig. 7. Comparison of simulations to observed planet positions for planet e, d, c, and b subsequently. The histograms show the star-planet distance in our system 94, simulation 4e, after having moved all output to the observed position angle of the planet, bootstrapped a data set, and added a normally distributed error to the simulation data. We show, in black, dark grey, and light grey lines, the observed on-sky star-planet distances from Currie et al.(2011) assuming the mean Earth-HR 8799 distance (39.4 pc) and adding 1 and 2 σ of this distance estimate (van Leeuwen 2007). This procedure is described in detail in Sect. 4.2. simulation 4e. Instead, we present our recipes on how to create For the 4:2:1 resonance we consider the angle similar systems. φ = λ 3λ + 2λ , (7) We also create simulation 2α from system 8, simulation 2, 1, 4:2:1 inner − middle outer using the same procedure as when creating simulation 4α. which results in two angles as there are two sets of three-planet Figure9 shows with time the number of systems in simula- systems; b-c-d and c-d-e. tions 2α and 4α that have not experienced any close encounter. For the 8:4:2:1 resonance we take into account the four an- The slopes of the curves show that the systems indeed have dif- gles as follows: ferent evolutions despite their identical initial conditions. The φ = λ 2λ λ + 2λ (8) location of the curves in the diagram shows that the stability 1, 8:4:2:1 e − d − c b timescale of simulation 4α is about one order of magnitude φ2, 8:4:2:1 = 2λe 5λd + λc + 2λb (9) longer than the stability timescale of simulation 2α. This leads to − φ , = λ λ 4λ + 4λ (10) 26 systems in simulation 4α surviving longer than the estimated 3 8:4:2:1 e − d − c b φ = λ 4λ + 5λ 2λ , (11) age of HR 8799 and 10 of these also matching the observations 4, 8:4:2:1 e − d c − b of HR 8799. which we now label with the planet names. We also look at the difference in longitude of periastron, 4.4. Mean motion resonance analysis which is an angle related to secular motion. It is defined as ϕ = $ $ , (12) In this section we investigate whether resonances affect our inner − outer systems in simulation 4. Stabilisation because the system which leads to three angles from the three planet pairs (b-c, c-d is deep in a mean-motion resonance is a common expla- and d-e). nation to the survival of the HR 8799 system (see e.g. In Fig. 10 we show the evolution of four of the above an- Fabrycky & Murray-Clay 2010; Go´zdziewski & Migaszewski gles in system 94, simulation 4e. We want to stress that sys- 2009, 2014; Reidemeister et al. 2009; Moro-Martín et al. 2010; tem 94, simulation 4e is a system similar to many of our Currie et al. 2011; Soummer et al. 2011; Esposito et al. 2013). other systems in the resonant behaviour seen in Fig. 10 also. In the following we show that we simulate systems that fit the The evolution of the angles in our panels 2, 3, and 4 in HR 8799 system without being stabilised by strong resonant Fig. 10 are also shown in the mean motion resonance analy- lock. sis of Go´zdziewski & Migaszewski(2009, their Figs. 4, mid- We consider resonant angles correlated to the Laplace reso- dle panel, and 6, top panel) and Go´zdziewski & Migaszewski nances, 2:1, 4:2:1, and 8:4:2:1 (Murray & Dermott 1999). We (2014, their Fig. 1, bottom right panel). The previously pre- also consider angles related to the secular motion. All these sented solutions (see e.g. Go´zdziewski & Migaszewski 2009, angles are calculated using the mean longitude (λ) and longi- 2014; Fabrycky & Murray-Clay 2010) all show small-amplitude tude of periastron ($), which are calculated from the longitude librations of one or more resonant angles. In contrast, our so- of ascending node (Ω), argument of periastron (ω), and mean lution shows at most transitions between large-amplitude libra- anomaly (M) for each planet in the following way: tion and circulation. This configuration does not guarantee the indefinite protection from close encounters that small-amplitude λ = Ω + ω + M (3) librations provide. $ = Ω + ω. (4) To show the overall behaviour of our simulation 4 we bin the systems by stability time into 9 bins, each containing 100 sys- The resonant angles we consider of the 2:1 resonance are tems. For each system we assess whether the resonant angles presented in Eqs. (5) (12) are circulating, librating, or falling in φ = 2λ λ $ (5) − 1, 2:1 outer − inner − outer and out of resonance. We visualise the resonant behaviour of our φ2, 2:1 = 2λouter λinner $inner, (6) simulation 4 in Fig. 11. Figure 11 shows no trend of longer sta- − − bility time with librating angles (right panel) and no clear trend which for the three planet pairs (b-c, c-d and d-e) result in six of stability time angles transitioning between librational and cir- angles to consider. culatory behaviour. We infer from this that our systems are not

A147, page 8 of 14 Y. Götberg et al.: Long-term stability of the HR 8799 planetary system without resonant lock

(a) Nominal mass (5, 7, 7 and 7 MJup) 100 d? = 39.4 pc Simulation 4 0 10 80 t > 30 Myr like HR 8799 60 like HR 8799, 10 1 − t > 30 Myr 40 Fraction of runs in bin Number of systems 20 Simulation 4α 2 Simulation 2α 10− (b) Nominal mass (5, 7, 7 and 7 M ) 0 Jup 103 104 105 106 107 108 d? = 39.4+1.1 pc Simulation 4 100 Time [yrs] Fig. 9. Comparison between simulations 4α (violet) and 2α (yellow). The lines show the number of systems in the simulation that have not experienced a close encounter, plotted with time. The slightly wider 1 α α 10− orbits in simulation 4 compared to simulation 2 makes the systems close encounter time shift about an order of magnitude. The black line and the grey zone show the age estimate and uncertainty of HR 8799. Fraction of runs in bin

2 10− 4.5. Simulations 5 and 6: lower planet masses (c) Lower mass (4, 6, 6 and 6 MJup) d? = 39.4 pc Simulation 5 Another way to increase the stability timescale of a planetary 100 system is to decrease the planet masses. In terms of ∆ (Eqs. (1) and (2)), lowering planet masses widens the orbit separations and thus increases the stability timescale. We use the exact same initial conditions for simulation 4 and lower the planet masses to create simulation 5 and 6 (see 1 10− Table1). Simulation 5 has planet masses 4 (b) and 6 (cde) MJup and simulation 6 has planet masses 3 (b) and 5 (cde) MJup. We

Fraction of runs in bin want to point out that these planet masses are still comfortably within the estimated masses from cooling models (5 2 M (b) ± Jup 2 +3 10− and 7 2 MJup (cde); Marois et al. 2008, 2010). 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 An− example of a system that fits from simulation 5 is system 24, simulation 5d seen in Figs. 12 and 13. The system does Binned ∆min [RH] not experience any close encounter during the 100 Myr inte- Fig. 8. We show how many of our systems in simulation 4 (top and gration time and we measure the quantiles to be qe = 0.93, middle panels) and simulation 5 (bottom panel) fit the observations. We qd = 0.98, qc = 0.15 and qb = 0.07 at the nominal Earth- binned the simulations in ranges of minimum orbit separations, ∆min, HR 8799 distance 39.4 pc. We consider this system to be a good 100 systems in each bin. The light blue shows the fraction of the sys- fit to the observations of HR 8799. The system is not stabilised tems in the bin that have a stability timescale longer than the estimated by low-amplitude librational behaviour. age of HR 8799. The dark blue curve shows the fraction of the systems in the bin that have architectures fitting to that of HR 8799. We deter- Using the technique described in Sect. 4.2 we find that for simulation 5, in the best case (where ∆ = 3.81 3.85) 18 sys- mine whether a system fits through the procedure described in Sect. 4.2. min − The grey bars indicate the fraction of systems in each bin that fulfil tems of 100 fit the observations. Looking at a broader range both criteria. The top panel shows the results from simulation 4 using of orbit separations in total 82 out of 900 systems fit the ob- the mean of the Earth-HR 8799 distance estimate (39.4 pc), the middle servations (see Fig.8c). Considering these 1 MJup lower planet panel shows the results from simulation 4 using a one standard devi- masses in simulation 5 the architecture of HR 8799 is becom- ation longer Earth-HR 8799 distance (40.5 pc), and the bottom panel ing common in our systems also after 30 Myr. The boost of fit- shows the results from simulation 5, which has slightly lower planet ting systems in simulation 5 is clearly seen in Fig.8 when com- masses, but is still within the estimated range. paring the bottom panel (simulation 5) with the top two panels (simulation 4). We include simulation 6 to prove the point that lower planet being stabilised by tight resonant lock. However, the separatrix- masses implies much longer stability timescale. In simulation 6 crossing behaviour we observe in some systems is characteris- we find 64 fitting systems of 100 in the best case (when ∆min = tic of chaotic evolution driven by multiple resonances (Chirikov 3.98 4.02) and 241 of 900 when considering all orbit separation − 1979). Indeed, this may offer an explanation as to why there is a ranges. long-lived tail of surviving systems with no consistent resonant Considering a slightly larger Earth-HR 8799 distance in- protection: Shikita et al.(2010) describe how some trajectories creased the number of fitting systems with a factor of two stick for extended periods at the edge of the chaotic region of (Sect. 4.1). Considering slightly lower planet masses boosts the phase space. number of fitting systems by a factor of ten.

A147, page 9 of 14 A&A 592, A147 (2016)

System 94, simulation 4e π

1c:2d 0 2 , φ π − 2π

π 1b:2c:4d 1 , φ 0

2π

π 1b:2c:4d:8e 1 , φ 0 π d

− 0 c π − 0 5 10 15 20 25 30 35 40 45 Time [Myr]

Fig. 10. Evolution of (from top to bottom) the resonant angles φ2, 1c:2d, φ1, 1b:2c:4d, φ1, 1b:2c:4d:8e and $c $d for system 94, simulation 4e. The black − dots show the output from our simulation, sampled every 100 years. In the top panel (evolution of the φ2, 1c:2d) the output clusters around 0 rad, which is a sign of weak resonant behaviour. The system alternates between libration and circulation in this resonant angle, which is a different behaviour than what is seen in systems affected by strong resonance in which all output would be narrowly confined about 0 rad. None of the bottom three panels show signs of resonant libration, unlike in the cases considered by Go´zdziewski & Migaszewski(2009, 2014). The red line indicates the estimated age of HR 8799.

Circulation Transitioning Libration 100 de 1 − 2 1 2 cd 90 − 1 2 bc − 80 φ4 bcde

φ3 bcde 70 N

φ2 bcde b r

60 o φ1 bcde f s y st e φ1 cde 50 m

φ bcd s 1 i 40 n b

φ de i

2 n

φ2 cd 30

φ2 bc 20 φ1 de

φ1 cd 10

φ1 bc 0 106 107 108 106 107 108 106 107 108 Stability time [yrs] Stability time [yrs] Stability time [yrs]

Fig. 11. Resonant behaviour of our simulation 4, binned with stability time (x-axis) and 100 systems per bin. Colour shows for diﬀerent resonant angles (y-axis) how many systems in the bin are circulating (left panel), librating (right panel) or transitioning between a resonant and non-resonant state (middle panel). We consider the angles in Eqs. (5) (12) (see Sect. 4.4). Our stability times in simulation 4 stretch between 0.04 Myr up to stable systems after 100 Myr. We see a resonant behaviour− in the 2:1 resonant angles where it is common that the systems transition between libration and circulation. However, there seems to be no dependence on stability time with resonances.

A147, page 10 of 14 Y. Götberg et al.: Long-term stability of the HR 8799 planetary system without resonant lock

System 24, simulation 5d is unlikely to observe a system in phase B unless the star is very young.

2 tce > 100 Myr 5.1. Application to observed wide-orbit planetary systems

1 In this section we predict the evolutionary phase of directly imaged, massive, wide-orbit planetary systems assuming their evo-

["] lution is similar to that which we ﬁnd for HR 8799 in simu- 0 lation 4f_long. Table2 gives some observed properties of the

Nort h systems we include. 1 − 1RXS J1609 1RXS J1609 b is found at a projected distance of 330 AU and is probably scattered to this distance as a pro- 2 − toplanetary disk of this size is unlikely. Its age is not suﬃcient to distinguish between phase B or C, but if there are no other 2 1 0 1 2 − − planets present we can predict phase C. East ["]

Fig. 12. System 24 in simulation 5d (using lower planet masses of AB Pic AB Pic b is also found at a very large distance from the 4 MJup for planet e and 6 MJup for planets b, c, and d). Black dots are star ( 260 AU). It is highly likely the planet has been scattered our output data points assuming the inclination and position angle of there∼ and also taking the age of 30 Myr into account, we predict 26◦ and 64◦ from Matthews et al.(2014). The coloured dots are the ob- phase C as phase B generally lasts∼ for shorter time. servations of the HR 8799 planetary system. The system matches the data and does not experience any close encounter during our 100 Myr integration time. β Pic β Pic b is separated from β Pic by 8 15 AU. The fact that no outer planets have been found indicates− either that the system 5. Evolutionary phases is a one-planet system in phase C or that the outer planets are not massive enough to be visible and the system is in phase A In this section we use simulation 4f_long (see Table1) to in- or C. However, the low eccentricity of the orbit (e 0.06, vestigate the evolution of wide-orbit planetary systems. Simula- Macintosh et al. 2014) favours phase A. ∼ tion 4f_long is created in the same way as simulation 4f, but the integrations are not stopped after a close encounter has occurred and therefore always continue up to 100 Myr. Fomalhaut Fomalhaut b is found at a projected distance of 119 AU, which is far out and makes in situ formation less prob- We find three distinct phases of dynamical evolution in our ∼ wide-orbit planetary systems. We call the phases A, B, and C able. The predicted eccentric orbit necessary to fit the observa- and they correspond to the time before the first close encounter tions would indicate phase C considering the advanced age of the system ( 440 Myr). between planets (A), the time between the first and last close ∼ encounter (B), and the time after the last close encounter (C). We define a close encounter between planets to be whenever two planets enter their mutual Hill radius, see Eq. (1). We can use GJ 504 The recently imaged planet around the solar-type star these evolutionary phases to predict what will happen and what GJ 504 (Kuzuhara et al. 2013) is found at a projected star-planet has happened to the HR 8799 system and systems similar to it. distance of 43.5 AU. GJ 504 b could be an outer planet in a phase C system (consider the age of 160 Myr), which means Figure 14 shows how the eccentricity and semi-major axes ∼ of the planetary orbits change between phase A, B, and C. In the that there might be another planet on a closer orbit. figure we picked one random time in phase A, B, and C in each system in simulation 4f_long, respectively. HD 106906 HD 106906 b has an on-sky projected separation Phase A is characterised by low-eccentricity orbits, not devi- of 650 AU, which makes it clear that the planet is scattered ating much from the initial set-up. Phase B and C show an even and∼ the system is in phase B or C as such a large protoplanetary distribution of eccentricities (0 < e < 1) and planets pile-up disk is unlikely. It may be that there is also an inner planet in the around a 10 AU. All systems in phase B have three planets HD 106906 system as the inner part cannot yet be resolved. or more, while∼ the most common number of planets in phase C is two planets. In the two-planet case of phase C, one planet typically has a semi-major axis of a 10 AU and the other HD 95086 ∼ HD 95086 b has a projected distance to the star of a 30 1000 AU. The timescale of phase A is what we referred . . ∼ − 56 4 0 7 AU and could be an outer planet in a phase B or C to as the stability timescale, which is the time it takes before the configuration,± which then indicates possible inner companions. first close encounter occurs. This timescale is strongly depen- It could also be a phase A system with an unresolved inner re- dent on orbital separation (Davies et al. 2014). The timescale of gion because of the distance to the star (90.4 3.4, van Leeuwen phase B is independent of initial orbit separation and we mea- 2007). The debris disk of HD 95086 is found± to be similar to that 4 7 6 sure it to vary between 10 to 10 years, peaking at 10 years. of HR 8799 (Su et al. 2015). These timescales for the clearing of wide-orbit planets are consistent with previous studies (Veras et al. 2009). This timescale is typically shorter than the duration of phase A and the lifetime HR 8799 The number of planets in HR 8799 and their low ec- of the observed systems (see Table2). Therefore statistically it centricities make us classify the system as still being in phase A.

A147, page 11 of 14 A&A 592, A147 (2016)

e d c b

d⋆ = 39.4 pc d⋆ = 39.4+1.1 pc d⋆ = 39.4+2.2 pc

10 12 14 16 18 18 20 22 24 26 28 30 34 36 38 40 42 44 46 65 70 75 80 85 On-sky projected separation [AU]

Fig. 13. Same as Fig.7 but for the system 24, simulation 5d (planet masses are lower than nominal, M = 4 MJup(b), and 6 MJup (cde)). Evolutionary phase A B C

0 % in A 0 % in B 13 % in C 1 0.8

1p 0.6 e 0.4 0.2 0 0 % in A 0 % in B 80 % in C 1 0.8

2p 0.6 e 0.4 0.2 0 0 % in A 45 % in B 7 % in C 1 0.8

3p 0.6 e 0.4 Number of planets 0.2 0 100 % in A 55 % in B 0 % in C 1 0.8

4p 0.6 e 0.4 0.2 0 1 10 100 1000 1 10 100 1000 1 10 100 1000 a [AU] a [AU] a [AU]

Fig. 14. Eccentricity and semi-major axes of the planets in each system of simulation 4f_long at a randomly picked time for each system in each phase. From top to bottom we show the cases of 1-, 2-, 3-, and 4-planet systems and from left to right we show phase A, B, and C. All phase A systems have low eccentricity and four planets. In phase B, the systems have three or more planets. Typically, systems in phase C have two planets, one at a 10 AU and one at 30 < a < 1000 AU. Both phase B and C show generally higher eccentricities. ∼ ∼ ∼

κ And The κ And system seems like a higher mass version of 6. Summary and conclusions the systems HD 95086 and GJ 504. The inner regions might hide a phase A system whilst the outer regions might reveal an outer The star HR 8799 is observed to be accompanied by four planet indicating phase C. The system is old enough for us to massive planets on wide orbits. Puzzlingly, the planetary sys- predict phase B to be unlikely. tem has been shown to be unstable in a timescale much

A147, page 12 of 14 Y. Götberg et al.: Long-term stability of the HR 8799 planetary system without resonant lock shorter than its estimated age. A common explanation of its relatively short for massive, wide-orbit planetary systems and it existence has been stabilisation by mean motion resonances thus makes sense that only one system has been observed in this (Go´zdziewski & Migaszewski 2014). phase, that is HR 8799. We have simulated long-lived systems similar to HR 8799 Acknowledgements. The authors thank the anonymous referees for their com- without forcing them into resonance. The initial orbits of ments which improved the manuscript. The authors thank Phil Uttley for ex- our planets are slightly more widely separated than circular pertise in statistics. Y.G. acknowledges the support of a Ph.D studentship at the orbits that intersect the observed star-planet on-sky separa- University of Amsterdam under the supervision of Selma E. de Mink. A.J. and tions. Slightly larger separations between the orbits result in a Y.G. were partially funded by the European Research Council under ERC Start- significantly longer stability timescale. The low eccentricity of ing Grant agreement 278675–PEBBLE2PLANET and by the Swedish Research Council (grant 2010 3710). MBD was supported by the Swedish Research the orbits can in some cases make a system have the observed Council (grant 2011-3991).− R.P.C. was supported by the Swedish Research architecture of HR 8799. We place the planets with random az- Council (grants 2012 2254 and 2012 5807). Calculations presented in this pa- imuth angle, assuming initial low eccentricities. per were carried out− at LUNARC using− computer hardware funded in part by When using the estimated planetary masses (5, 7, 7 and the Royal Fysiographic Society of Lund. A.M. was supported by the Swedish Research Council (grant 2011 3991) and KAW 2012.0150 from the Knut and 7 MJup) together with the estimated Earth-HR 8799 distance we Alice Wallenberg foundation. − rarely find a fitting system; in the best case, ∆min = 3.64 3.67, we find 5 of 100 fitting systems, and in the broader range of− orbit separations we find 12 out of 900 fitting systems. We almost double the number of fitting systems by considering the 1σ longer References distance to HR 8799 of 40.5 pc; the best case now considers or- Bailey, V., Meshkat, T., Reiter, M., et al. 2014, ApJ, 780, L4 bit separation range ∆min = 3.74 3.79 and yields 8 of 100 fitting Baraffe, I., Chabrier, G., Barman, T. S., Allard, F., & Hauschildt, P. H. 2003, systems: in total we find 22 of 900− fitting systems. It is easier to A&A, 402, 701 find a fitting system by considering slightly lower planet masses Bowler, B. P., Liu, M. C., Kraus, A. L., & Mann, A. W. 2014, ApJ, 784, 65 Carson, J., Thalmann, C., Janson, M., et al. 2013, ApJ, 763, L32 (4, 6, 6, and 6 MJup) that are still comfortably within the planet Chambers, J. E. 1999, MNRAS, 304, 793 mass estimate. That results in the best case (∆min = 3.98 4.02) Chambers, J. E., Wetherill, G. W., & Boss, A. P. 1996, Icarus, 119, 261 18 of 100 systems fit the observations and in the broader− range Chatterjee, S., Ford, E. B., Matsumura, S., & Rasio, F. A. 2008, ApJ, 686, 580 of orbit separations we find 82 out of 900 systems fitting. The Chauvin, G., Lagrange, A.-M., Zuckerman, B., et al. 2005, A&A, 438, L29 above described numbers are visualised in Fig.8. Examples of Chirikov, B. V. 1979, Phys. Rep., 52, 263 Currie, T., Burrows, A., Itoh, Y., et al. 2011, ApJ, 729, 128 a system that fits the observations can be seen for the nominal Davies, M. B., Adams, F. C., Armitage, P., et al. 2014, Protostars and Planets VI, planet masses in Figs. 5b,6, and7 and for the slightly lower 787 planet masses in Figs. 12 and 13. Esposito, S., Mesa, D., Skemer, A., et al. 2013, A&A, 549, A52 We test our systems for mean motion resonances (see Fabrycky, D. C., & Murray-Clay, R. A. 2010, ApJ, 710, 1408 Galicher, R., Rameau, J., Bonnefoy, M., et al. 2014, A&A, 565, L4 Sect. 4.4) and find that most of them fall sporadically in and Go´zdziewski, K., & Migaszewski, C. 2009, MNRAS, 397, L16 out of the 2:1 resonance between various planet pairs, however, Go´zdziewski, K., & Migaszewski, C. 2014, MNRAS, 440, 3140 we see no trend in long stability timescale in the systems with Gray, R. O., & Kaye, A. B. 1999, AJ, 118, 2993 more resonant behaviour. This means that these resonances do Gray, R. O., Corbally, C. J., Garrison, R. F., McFadden, M. T., & Robinson, P. E. 2003, AJ, 126, 2048 not work in a stabilising way, unlike the resonances treated ear- Johansen, A., Davies, M. B., Church, R. P., & Holmelin, V. 2012, ApJ, 758, 39 lier in the literature for stability of the HR 8799 system. Kalas, P., Graham, J. R., Chiang, E., et al. 2008, Science, 322, 1345 We conclude that the HR 8799 system does not necessarily Kennedy, G. M., & Wyatt, M. C. 2011, MNRAS, 412, 2137 have to be caught in a strong resonance. We simulated many sys- Kenyon, S. J., Currie, T., & Bromley, B. C. 2014, ApJ, 786, 70 tems that at some point in their evolution look like HR 8799 and Kuzuhara, M., Tamura, M., Kudo, T., et al. 2013, ApJ, 774, 11 Lafrenière, D., Jayawardhana, R., & van Kerkwijk, M. H. 2010, ApJ, 719, 497 we want to stress that our example system (system 94, simula- Lagrange, A.-M., Bonnefoy, M., Chauvin, G., et al. 2010, Science, 329, 57 tion 4e) is not a lucky coincidence. 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After the first close encounter has occurred, the first Murray, C. D., & Dermott, S. F. 1999, Solar System Dynamics (Cambridge 6 planet ejection follows within 10 years. The close encounter University Press) and ejection change the architecture∼ of the system and the eccen- Pecaut, M. J., Mamajek, E. E., & Bubar, E. J. 2012, ApJ, 746, 154 tricities are now more evenly distributed (0 < e < 1). In 80% Perryman, M. A. C., de Boer, K. S., Gilmore, G., et al. 2001, A&A, 369, 339 ∼ Pueyo, L., Soummer, R., Hoffmann, J., et al. 2015, ApJ, 803, 31 of the cases, the systems eventually lose another planet and end Rameau, J., Chauvin, G., Lagrange, A.-M., et al. 2013, ApJ, 772, L15 up with two planets on eccentric orbits: one with a semimajor Reidemeister, M., Krivov, A. V., Schmidt, T. O. B., et al. 2009, A&A, 503, 247 axis of ainner 10 AU and the other at aouter 30 1000 AU (see Shikita, B., Koyama, H., & Yamada, S. 2010, ApJ, 712, 819 Fig. 14). 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A. 2001, ApJ, 562, L87 simply in an earlier evolutionary phase. This phase seems to be

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Appendix A: Initial conditions of system 94, simulation 4e Table A.1. Initial conditions of system 94, simulation 4e.

Planet b c d e Mass [M ] 0.004775549 0.006685769 0.006685769 0.006685769

x [AU] 72.3056194919 40.2180215763 13.0773477594 14.2619630629 y [AU] −18.4404318999 −17.5354438553− 21.3633582260 0.0317906084 z [AU] − 0.4918207713− 0.1521487580 0.0232168947 1.0418248166 − −

vx [AU / day] 0.0006026600 0.0012707737 0.0035903173 0.0000263986 v [AU / day] 0.0023632584 0.0029135670 −0.0021977681− 0.0055679781 y − − − vz [AU / day] 0.0000077237 0.0001137888 0.0000052188 0.0001914785

Notes. This planetary system is so chaotic that these initial conditions are unlikely to reproduce our system 94, simulation 4e. A small change in timestep could alter the lifetime drastically.

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